Answer:
9 and 10
Explanation:
x = 1st positive integer
x+1 = 2nd positive integer
Solve for both x and x+1
(x+1) is the next positive integer after x (Ex. 3 aka 2+1 is next integer after 2).
Larger Integer: x+1 ==> being squared
Smaller Integer: x ==> being multipled by 9
9x + 19 = (x+1)^2 ==> the square of (x+1) is 19 more that 9 multiplied by x.
9x + 19 = (x+1)(x+1) ==> simplify
9x + 19 = x(x+1) + 1(x+1) ==> distribute (x+1) to x and 1.
9x + 19 = (x*x + 1*x) + (1*x + 1*1)
9x + 19 = x^2 + x + x + 1 ==> simplify
9x + 19 = x^2 + 2x + 1
19 = x^2 - 7x + 1 ==> subtract 9x on both sides
Make one side of the equation equal to 0 so you can factor the equation:
0 = x^2 - 7x - 18 ==> subtract both sides by 19
0 = x^2 + 2x - 9x - 18 ==> 2x - 9x = -7x and 2 * (-9) = -18
0 = x*x + 2x - 9x - 18
0 = x(x + 2) - 9(x + 2) ==> factor
0 = (x - 9)(x + 2)
x - 9 = 0 x + 2 = 0
x = 9 x = -2
Out of x = 9 and x = -2, only x = 9 is positive.
Hence, x = 9 and x+1=10.
Hence, the answers are 9 and 10.